Here, we consider the two useful Anderson-Grüneisen
parameters
in more detail [*Grüneisen,* 1926;
* O. Anderson,* 1966a, 1967]:
the
isothermal
*d*_{T} parameter introduced above
(see (31)) and
the adiabatic
*d*_{S} parameter defined as

(82) |

Both parameters are used in geophysical and physical studies
[*Chung,* 1973; * Barron,* 1979].
The parameter
*d*_{T} is
largely applied in analyzing the
*P*-*T* behavior of
*a*,
*K*_{T}, and
*t*, and
*d*_{S} is used to estimate the
temperature dependence of
*K*_{S} and to treat the relations
between elastic properties (elastic wave velocities) and
thermodynamic data
[*D. Anderson,* 1987;
* O. Anderson et al.,* 1987;
* Isaak et al.,* 1992;
* Duffy and Ahrens,* 1992a, 1992b;
* Agnon and Bukowinski,* 1990b].

As the temperature decreases in the range
*T* 300 K at
*P* = constant, both parameters
*d*_{S} and
*d*_{T} sharply increase due to decreasing
*a*, but at high
temperatures,
*T* > *Q*, they become more or less constant
[*O. Anderson et al.,* 1992a]. From (16), we derive
the identity
relating
*d*_{T} and
*d*_{S} [*Birch,* 1952]

(83) |

which was used to calculate the
*d*_{T} values listed in
Table 3 and * Pankov et al.* [1997]. Deriving (83),
we find in
passing that

(84) |

where, for convenience, the notation

(85) |

is introduced.

The values of
*L* and
(*g*/ *T*)_{P} at
*P* = 0,
calculated by (84) and (85),
are also presented in Table 3 and * Pankov et al.* [1997].

Data for some minerals [*O. Anderson et al.,* 1992a]
show that
(*g*/ *T*)_{P=0}
0 over a wide temperature range. Assuming that
(*g*/ *T*)_{P} = 0 for
*C*_{V} = constant
(*T* > *Q*), (84) gives

(86) |

Moreover, since
*C*_{V} = constant, this case results in
*q* = 0,
and consequently, according to (35) and (42),

(87) |

However, the estimation of
*d*_{T} by (42) for
*q* 1 is more
accurate than the values from (87). Then, one might expect that
in (84)

for
*C*_{V} constant.

Using (83), identity (84) can be rearranged to the form

(88) |

If the last term in (88) is small, then
[*O. Anderson et al.,* 1992a]

(89) |

This approximation is recommended for evaluating of the Anderson-Grüneisen parameters at high temperatures, when there are no sufficient data for applying (83) or (88).

In addition to the analysis of the parameter
*d*_{T}(*P*, *T*) described
in sections 6 and 8, we consider the following
features in the behavior of
*d*_{T}. 1) If
*g* = *g*(*V*),
then
*q* = *q*(*V*), but generally speaking,
*d*_{T} = *d*_{T}(*V*,
*T*) since
*C*_{V} = *C*_{V}(*V*, *T*) and
*K*^{} = *K*^{}(*V*, *T*). 2) If
*C*_{V} = *C*_{V}(*T*),
then (42) holds true, and moreover,
*g* = *f*(*V*)/*C*_{V}(*T*),
*q* = *q*(*V*),
although, generally speaking,
*K*^{} = *K*^{}(*V*, *T*) and
*d*_{T} = *d*_{T}(*V*,
*T*). Combining the former of these assumptions
with the condition
*K*_{T} = *K*_{T}(*V*) (i.e.,
*d*_{T} = *K*^{} ), we
find

(90) |

and
*C*_{V} therefore takes the form

In section 6.2.2, the arguments were given for decreasing
*d*_{T} with pressure. The same behavior
of this parameter
follows from the approximation (42) since both
*K*^{} and
*q* decrease with pressure. The exact relations for the
*P* and
*T* derivatives of
*d*_{T} result from the definition
of
*d*_{T} by (31)

(91) |

(92) |

If the first term in (91) prevails, we have an unusual case
((*d*_{T})/( *P*))_{T} > 0.
Neglecting the second temperature derivative in (92) (at
least at room temperature), we find that
(*d*_{T}/ *T*)_{P} < 0.
However, the approximation
*d*_{T} = *K*^{} (more realistic at
*T* > *Q* )
gives, by constast,
(*d*_{T}/ *T*)_{P} > 0 because of
( *K*^{}/ *T*)_{P} > 0. Experimental
data
indicate that
((^{2} *K*_{T})/( *T*^{2}))_{P} for
*T* > *Q* is negative and small in value
[*O. Anderson et al.,* 1992a].

Finally, the EOS of type (28), which we used to calculate the
thermal expansion coefficient by (36), allows us to determine
the explicit pressure (or volume) dependence of
*K*_{T} along an
isotherm. Retaining in (36) only the terms involving
*dK*^{}_{0}/*dT*,
we obtain

(93) |

where
*f*^{} denote the derivative of
*f* with respect to
*x*. This
approximation generalizes the similar Birch formula that follows
from (93) for
*dK*^{}_{0}/*dT* =
0 [*Birch,* 1968].
However, when using usual EOS
types,
*d*_{T} from the Birch formula increases
with
*P* (except for the Murnaghan EOS for which the behavior of
*d*_{T} depends on the sign of the
difference
*K*^{} - *d*_{T0} ).
Again, we convince ourselves that the term with
derivative
*dK*^{}_{0}/*dT* is
important in analyzing
the thermal expansion by (28).

Figure 8 shows several curves of
*d*_{T}(*x*) calculated using
(93) and EOS (37), which correspond to the curves of
*a* in
Figures 1 and
3 (the straight line
*d*_{T} = 6*x* - 1 by (47) is
also drawn for comparison). The favored value of
*dK*^{}_{0}/*dT* is
2 10^{-3} K
^{-1} (see section 6.2), and deviations from it
substantially affect the
*d*_{T} values at compressions in
the lower mantle. At high temperatures, according to (42), we
have also the lower limit for
*d*_{T},
*d*_{T} > *K*^{} - 1 [*O. Anderson et al.,* 1992a].

Elastic moduli and their
*P*-*V*-*T* derivatives are the
characteristics constituting the basis of the Earth's interior
thermodynamics [*Birch,* 1952, 1961;
* Sumino and O. Anderson,* 1984;
* D. Anderson,* 1967, 1987, 1989;
* O. Anderson et al.,* 1992a;
* Duffy and D. Anderson,* 1989;
* Bina and Helffrich,* 1992;
* Duffy and Ahrens,* 1992a, 1992b].
These quantities also serve as
parameters of EOS's. The simplest estimates of adiabatic and
isothermal bulk moduli at high pressure are given by their
linear pressure dependences, which, however, begin to
overestimate the bulk modulus at a compression of about
*x* < 0.85.
The
*P*-*T* variation of the pressure derivative was
considered in many papers devoted to EOS's (see section 3). In
order to assess the applicability of empirical EOS's, one
often uses a relation between the first
*K*^{}( *K*_{T}/ *R*)_{T} and second
*K*^{} (^{2}*K*_{T}/
*P*^{2})_{T}
pressure derivatives at
*P* = 0 [*Pankov and Ullmann,* 1979a;
* Jeanloz,* 1989;
* O. Anderson,* 1986;
* Hofmeister,* 1991b]. Values of
*K*^{} and
*KK*^{}
at
*P* = 0 generally lie in the intervals 4-6 and
-(5-10), respectively. The uncertainty in
( *K*_{S}/ *P*)_{T} (measured by ultrasonic or Brillouin
scattering methods) can reach 1-5%
(with allowance for data from various laboratories) or, in some
cases, 20% and even 50%. Anomalous values of
*K*^{} and
*K*^{}
are sometimes reported (see references in tables of
* Pankov et al.* [1997]): for example,
*K*^{} = 5-7 (garnet),
*K*^{} = 9-14 (pyroxene),
and
*KK*^{}
= -60 (spinel), which are assumed to take
on more usual values as pressure increases.

Let us turn our attention to the relations between the
adiabatic-isothermal derivatives of
*K*_{S} and
*K*_{T}. Changing the
variables
*P* and
*S* to
*P* and
*T*, we find

(94) |

where
*d*_{S} is defined by (82). The derivative
( *K*_{S}/ *P*)_{S} characterizes the curvature of an adiabatic
*P*-*V* or a Hugoniot
curve. Further, from (16)

(95) |

Eliminating
(*g*/ *P*)_{T} with the help of (8) and (17) (or (122))
and using
(58), we find

(96) |

Substituting (94) for
( *K*_{S}/ *P*)_{T} and (93) for
*d*_{S}, we arrive at the * Birch*
[1952] formula

(97) |

The difference between the adiabatic-isothermal derivatives of
*K*_{S} and
*K*_{T} at normal conditions are generally small (1-2%
for mantle minerals). Data and estimations by (94)-(97) show
that we usually have

(except for FeO for which
( *K*_{S}/ *P*)_{T} is poorly known [*Pankov et al.,*
1997]);
however, * D. Anderson* [1989] indicated the inverse
inequality

For our high-temperature estimates given in Table 3 and * Pankov et al.*
[1997], it was arbitrarily assumed that

Hence, using (94)-(97), we found substantial differences (up to
10-30%) between the pressure derivatives of
*K*_{S} and
*K*_{T} at
high temperatures. In fact, the differences are of the same
order of magnitude as the derivative increments due to
increasing temperature. Specifically, the estimated
^{2}*K*_{T}/ *P*
*T* values are
3.5 10^{-4} (stishovite),
2 10^{-4} (ilmenite),
3 10^{-4} (Mg-perovskite), and
1.7 10^{-4} (MgO)
and do not exceed
1 10^{-4} for other minerals
(although some estimates appear to be negative).

* Isaak* [1993] estimated
^{2}*K*_{T}/ *P*
*T* using an identity of type (91) and
Boehler's data on the adiabatic temperature gradient (see also
section 12.2). He found
^{2} *K*_{T}/ *P*
*T* = (3.9 1.0)
10^{-4} and
(3.3 0.9) 10^{-4}
K
^{-1} for MgO and olivine, respectively.
Furthermore, he showed this derivative to decrease 30% as the
pressure increases isothermally to 100 GPa. A similar order of
magnitude was found from shock wave data to be a lower limit
for this derivative value [*Duffy and Ahrens,* 1992a] (see
also
sections 6 and 9).

In addition to the analysis of the mixed derivatives, we give the following identity

(98) |

which we derived from (94), using (127) and
(*g*/ *T*)_{P} from (88). Note
that a similar relation of * Bukowinski and Wolf* [1990]
is
different from (98) (because of either a reprint or mistake). To
give an example, we substitute in (98) the values typical of
the lower mantle:
( *K*_{S}/ *P*)_{S} = 4,
*a**g**T* =
0.1,
(( ln*d*_{S})/( ln*r*))_{S}
= -1,
and
*d*_{S} = 3 for
*x* = 1 and
*d*_{S} 2 for
*x* = 0.7. Then,

which is in agreement with the preceding estimates.

When considering the temperature behavior of
*K*_{S} and
*K*_{T}, the
Anderson-Grüneisen parameters
*d*_{S} and
*d*_{T} are
represented in the form of (29)
[*D. Anderson,* 1987;
* Duffy and D. Anderson,* 1989;
* O. Anderson et al.,* 1992a], which can be
rewritten as

(99) |

(100) |

In section 9, we considered the principal regularites in the
variation of
*d*_{S} and
*d*_{T} with pressure and
temperature (some decrease of them with
*T* in the vicinity of
*T* = 300 K,
the trend to constant values at
*T* > *Q*, and the
decrease with pressure). Now we dwell on the contributions of
the intrinsic
*d*_{V}^{S} and
*d*_{V}^{T} and extrinsic
*K*^{} anharmonic terms in (99)
and (100).

* D. Anderson* [1988, 1989]
pointed out that the temperature
variation of the bulk modulus at
*P* = constant mostly occurs by
the variation in
*a* ; i.e., here, the extrinsic
anharmonicity generally prevails and enhances with temperature
(due to increasing
*K*^{} ). The estimates given
in Table 3 and
* Pankov et al.* [1997] show that, at
*T* = 300 K, we have
*d*_{V}^{T} < 0 (except
very uncertain data for FeO); most of
the estimates falls into an interval between
-1 and
-2 (although it was found
*d*_{V}^{T} = -17,
-5, and
-3.9 for
coesite, stishovite, and fictive Fe-perovskite phase,
respectively). According to * D. Anderson* [1989],
the values of
*d*_{V}^{T} are typically
between
-4 and
-1 (his table 5,
however, contains values outside this interval: 2.2 for
orthopyroxene,
-5.3 for SrTiO
_{3}, and
-19 for CaCO
_{3} ). The
parameter
*d*_{V}^{S} satisfies
the inequality
|*d*_{V}^{S}| > 2 for
11 out of 54 minerals considered by D. Anderson
(specifically,
*d*_{V}^{S} = -4.1
(GeO
_{2} ), 3.8 (orthopyroxene),
-3.1 (SrTiO
_{3} ), and
-18 (CaCO
_{3} ). At high temperatures,
*d*_{V}^{T} can be
either positive or negative (between
-1 and
1), and
|*d*_{V}^{S}| is generally
positive, lying in the
interval 0-1.5 [*Pankov et al.,* 1997]. Thus, as temperature
increases,
|*d*_{V}^{T}|, on average,
decreases, but
|*d*_{V}^{S}| increases.
The contribution of
*d*_{V}^{S}/*d*_{S}
to (99)
is usually less than 10-30% at 300 K and does not exceed
15-60% at high temperatures. Accordingly,
*d*_{V}^{T} contributes
no more than 30-40% in (100) at 300 K and
usually less than 10% at high temperatures. From this analysis,
it follows that the extrinsic anharmonic term, although it
generally dominates in the derivatives of
*K*_{S} and
*K*_{T},
decreases its contribution in the case of
*K*_{S} and increases its
contribution in the case of
*K*_{T}. The decrease of
*d*_{V}^{T} with temperature
leads to the approximation

(101) |

and additionally, in view of (89) and (99),

(102) |

Thus, at high temperatures, namely the isothermal, rather than adiabatic bulk modulus becomes depending mostly on volume (i.e., temperature-independent function).

In his analysis of the thermodynamic properties of the lower
mantle, * D. Anderson* [1987, 1988,
1989]
relies on seismic
tomography and geoid data and assumes that the observed
horizontal velocity anomalies are caused by temperature
variations. * Stacey* [1992] showed, however, that it
is not
possible to explain the anomalies with a purely temperature
effect, since in such a case, the geoid highs would be too
great. Other hypotheses proposed to interpret the seismic
anomalies were related to inhomogeneities of composiion, or
partial melting, or even the presence of small amounts of fluids
[*Price et al.,* 1989;
* Duffy and Ahrens,* 1992b; and others].
Nevertheless, following D. Anderson, below, we estimate the
thermodynamic parameters for the lower mantle, considering the
temperature effect formally as a limiting case.

Based on the PREM model, the formula for the acoustic
Grüneisen parameter, and tomography data,
we have in the lower
mantle
( *K*_{S}/ *P*)_{S} = 3-3.8,
*g* = 1.2 0.1, and
*d*_{S} = 1-1.8.
Consequently, using (89), (94), (96), (99), and (100),
we find
*d*_{T}
*d*_{S} + *g*
= 2.2-3.0,
*K*^{} = 3-3.8 (a small correction
can be introduced with the help of
*a* = *a*_{0}*x*^{dT} for
*a*_{0} = 4
10^{-5} K
^{-1} and
*T* 2000-3000 K),
*d*_{V}^{S} *K*^{} - *d*_{S}
= 1.2-2.8, and
*d*_{V}^{T} = *d*_{V}^{S}- *g*
= -0.1-1.7.
If a greater uncertainty is assumed for
*g*, say
0.4, then
*d*_{T}
1.9-3.3 and
*d*^{S}_{V} -0.4-2.0.
Thus, although the extrinsic anharmonic effects weaken with
pressure, they still prevail under the lower mantle conditions
( *K*^{} > *d*^{S}_{V}
or
|*d*^{T}_{V}| ).
The intrinsic anharmonic term
*d*_{V}^{S} significantly
increases with pressure, but its isothermal analog
*d*_{V}^{T} can either
increase or decrease and reach zero.
D. Anderson, by reference to experimental data, points out the
case of
*d*_{V}^{T} 0, which yields
*d*_{V}^{S} *g* = 1.2.

However, this value of
*d*^{S}_{V} disagrees
with data for such a
representative lower-mantle material as periclase. Using
*d*_{V}^{T} = 0 and
*d*_{T} = *K*^{} 3.2-3.5 for
*x* = 0.7 and the Birch-Murnaghan EOS for MgO, we find
*d*_{S}
*d*_{T}- *g*
= 2.2-2.5 for
*g* = *g*_{0}*x*
( *q* = 1,
see (119)). These values of
*d*_{T} and
*d*_{S} are, on average, still exceed
the results inferred from seismic
models. * O. Anderson et al.* [1992b]
noted that data for MgO can be
reconciled with seismic results by assuming that
*q* < 1. In
particular, our analysis leads to the following consistent
sequence of values:
*d*_{V}^{T} 0.2,
*q* 0.8,
*g* = *g*_{0}*x*^{q}
1.1 (for
*x* = 0.7 ),
*d*_{T} = *K*^{}- *d*_{V}^{T}
3-3.3, and
*d*_{S} 1.9-2.2.
In any case, the consistency to seismic data could be found in this manner if
the values derived from seismic tomography were explained by
only horizontal variations of temperature.

The consideration of the temperature behavior of
*K*_{S} at
*P* = 0 will be added by the following two methods. One of
them uses the condition
*d*_{S} = *d*^{}_{S} = constant
[*O. Anderson,* 1988;
* Duffy and D. Anderson,* 1989], and in
view of (99), yields the power law

(103) |

which we used to estimate the values listed in Table 2 and
* Pankov et al.* [1997].

Another approach proposed by * O. Anderson* [1989]
and extended by
* O. Anderson et al.* [1992a] is based on data for
enthalpy. We
obtain the relation of
*K*_{S} to enthalpy using a somewhat
different procedure, namely, the formula
*g* = *a**K*_{S}*V*/*C*_{P}
from which the derivative
( *K*_{S}/ *H*)_{P} is found, and thus, approximately,

(104) |

where the asterisk marks the values at a fixed temperature. By
using the parameter values from Table 2
and * Pankov et al.* [1997],
as well as data for enthalpy, we estimated
*K*_{S} by
(104) for a number of minerals (Table 4). One can see that the
O. Anderson's method is quite efficient: the
uncertainty of the estimated values at high temperatures is
2-5%. It is also clear that both methods described above would
give more accurate results when high-temperature values for
*K*_{S}^{},
*r*^{},
*g*^{},
*H*^{}, and
*d*_{S}^{} are used in
the respective formulas.

In conclusion to this analysis, we show the dependence
of
*K*_{S} versus
*T* at
*P* = 0 (Figure 9) calculated by the
Mie-Grüneisen EOS with
*g* = *g*_{0}*x*^{q}
for three
minerals considered in sections 6-8. Comparing the
*K*_{S} curves for various values of
*q* = 0-2 with experimental data,
we see that it is possible to choose appropriate values of
*q* consistent to the data. However, considering the results
presented for the same EOS's in sections 6-8, it is not
always possible to find such values of
*q* for which the EOS
becomes consistent to data simultaneously for
*a*,
*C*_{P},
*t*, and
*K*_{S}. Thus, as in sections 6-8, we conclude that
the thermal part of the Mie-Grüneisen EOS
does not provide
suffuciently high accuracy of all the thermodynamic parameters
calculated from this EOS.